Some Mathematical Methods for Neurosciences

Présentation du cours 2017-2018 (avec E. Tanré)

Master M2 UPMC

Master M2 MVA

Stages Master - Biblio. / Refs. - Cours / Lectures - Where / When - Lecture Notes


We present a number of mathematical tools that are central to modeling in neuroscience. The prerequisites to the course are a good knowledge of differential calculus and probability theory from the viewpoint of measure theory. The thrust of the lectures is to show the applicability to neuroscience of the mathematical concepts without giving up mathematical rigor. The concepts presented in the lectures will be illustrated by exercise sessions.

  • Introduction to dynamical systems: orbits and phase portraits, invariant manifolds, equivalence of dynamic systems, topological classification of equilibria, structural stability, center manifold in finite dimension.
  • Introduction to bifurcation theory: dimension 1 (saddle-node, transcritical, pitchfork), dimension 2 (Hopf), center manifold, normal form, equivariant bifurcations.
  • Applications: ring model of orientations, Turing mechanism for cortical pattern formation, geometric visual hallucinations.
  • Mesoscopic models of visual cortical areas: anatomical structure of the visual cortex (V1), functional architecture of V1, neural fields models.
  • Neuronal models: aspatial Hodgkin-Huxley model, simplified models, synaptic models, spatial models.
  • Importance of noise: Brownian motion, stochastic differential equations, application to neurons.

Résumé


Nous présentons dans ce cours quelques outils mathématiques qui interviennent de manière systématique dans de nombreux problèmes de modélisation en neurosciences. Les prérequis sont une bonne connaissance du calcul différentiel et du calcul des probabilités dans le cadre de la théorie de la mesure. Sans trahir la rigueur mathématique, le cours s'efforcera de mettre en valeur l'applicabilité aux neurosciences des concepts présentés. Le cours sera complété par des séances d'exercices.

  • Introduction aux systèmes dynamiques: orbites et portraits de phases, variétés invariantes, équivalence de systèmes dynamiques, classification topologique des équilibres, stabilité structurelle, variété centrale en dimension finie.
  • Introduction à la théorie des bifurcations: dimension 1 (noeud-selle, transcritique, fourche), dimension 2 (Hopf), variété centrale, forme normale, bifurcations équivariantes.

Applications:

  • Modèles mésoscopiques de certaines structures corticales: structure anatomique du cortex visuel (aire V1), architecture fonctionnelle de V1, modèles de champs neuronaux.
  • Sensibilité à l'orientation des contours visuels, formation de structures corticales et hallucinations visuelles.
  • Modèles de neurones: le modèle de Hodgkin-Huxley sans espace, modèles simpliés, modèles de synapses, modèles spatiaux.
  • Le rôle du bruit: mouvement Brownien, équations différentielles stochastiques, application aux neurones.

Interships Master


Here are the projects proposed this year (more to come):

  • Jobs

Bibliographie sommaire (A few references)


  • Kandel, Eric R., éd. Principles of neural science. 5th ed. New York: McGraw-Hill, 2013.
  • Byrne, John H., Ruth Heidelberger, et Melvin Neal Waxham, From molecules to networks: an introduction to cellular and molecular neuroscience, 2014.
  • Gerstner, Wulfram, Werner M. Kistler, Richard Naud, et Liam Paninski. Neuronal dynamics: from single neurons to networks and models of cognition. Cambridge University Press, 2014.
  • Koch, Christof. Biophysics of Computation: Information Processing in Single Neurons. Oxford Univ. Press, 2004.
  • Bressloff, Paul C. Waves in Neural Media, Springer, 2014.
  • Eugène Izhikevich, Dynamical systems in neuroscience: the geometry of excitability and bursting, MIT Press, 2006.
  • G. Bard Ermentrout and David H. Terman, Mathematical Foundations of Neuroscience, Springer, 2010.
  • Sterratt, David, Principles of computational modelling in neuroscience. Cambridge University Press, 2011.

  • Yuri A. Kuznetsov, Elements of applied bifurcation theory.
  • Haragus, Mariana, et Gerard Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer London, 2011. http://link.springer.com/10.1007/978-0-85729-112-7.
  • Jean-Pierre Françoise, Oscillations en biologie, Springer, 2000.
  • Lawrence C. Evans, An introduction to stochastic differential equations, [Link]
  • Jean-François Le Gall, Mouvement brownien, martingales et calcul stochastique, 2013. [Link]
  • Sylvie Benzoni, Cours de M1 sur les EDOs, [Link]

Date et lieu des cours et des TPs ( When and where)


Location: Campus Jussieu : couloir 15/25, salle 103.

Les cours ont lieu les jeudis de 13:30 à 16:30, les séances de TDs de 16h45-18h45

The will be given at Jussieu from 13:30 to 16:30, tutorials: 16h45-18h45.

26 October (Slides)


  • Introduction to the Central Nervous System CNS
  • Models of a single neuron ( Hodgkin-Huxley )
  • Simplified models of spiking neurons
  • Poisson processes
  • Simulation Methods of Classical Laws
  • On the convergence rate of Monte Carlo methods

9 November (Slides)


This lecture starts 13:30 as scheduled.

  • Basics of systems dynamics (existence theorem, stability)
  • Introduction to planar models of single neurons (Morris–Lecar , FitzHugh-Nagumo, Integrate and Fire, Exponential Integrate and Fire...)
  • Introduction to local bifurcation theory (center manifold, codim 1)

The proof of the existence of the stable/unstable manifold is proposed in the exercises.

Exercises

It is important to do the persistence of limit cycles for the Hopf bifurcation.

16 November


Exceptionally, there are two lectures during the day. The first one is scheduled during 9:00-12:00, room 15/16-201 (Slides)

 

  • Normal form reduction.
  • Examples of bifurcations in neural models.
  • Introduction to delayed models in neurosciences.

The second lecture starts at 13:30 as usual in room 15/25-103 (Slides)

  • Biology of visual area V1 
  • Mathematics of Neural Fields (NF) 
  • Bifurcations in NF 
  • The Ring Model of orientation tuning

Exercises

23 November


Exceptionally, there are two lectures during the day. The first one is scheduled during 9:00-12:00, room 15/16-201

30 November



TBA

21 December


  • Biology of visual area V1 
  • Mathematics of Neural Fields (NF) 
  • Bifurcations in NF 
  • The Ring Model of orientation tuning

Examen


Location: Campus Jussieu

Time: 2pm to 5pm

Room: 15/25 - room 101

Problème d'examen :
Final examination:

4 pages manuscrites autorisées pour l'examen
You can bring 4 hand-written pages for the exam 

 

Examens années précédentes: 2015 - 2014 - 2013